A complex structure on S^6 [pdf](alpo.ge) |
A complex structure on S^6 [pdf](alpo.ge) |
Background: https://mathoverflow.net/questions/1973/is-there-a-complex-s...
complex structure in S^6 would mean an associative version of octonions (which are non associative "almost complex" versions of their cousins native to S^3, which you may know as (unit) "quaternions").
Also fancy cousins of unit vectors (that you can do a nonzero triple cross product with), ie "non associative vectors" living on S^2, the 2D surface of a sphere "with radius 1"
If you haven't quite got the PhD https://ncatlab.org/nlab/show/quaternionic+structure
Big enough that Atiyah the godfather of this subfield, in 2016 claimed to have solved it (not sure if it was before or after the dementia)
This is not just any math PhD, it's the anthropic guy whom Claude helped find the Jacobian counterexample, which might be quite worthy of a fields medal (for impact if not notoriety or complexity) if it were human
Disclaimer: not a pure maths PhD myself, just someone who likes this stuff so..
By the end of the first sentence I'm pretty sure it would take me quite some time to understand what it's about (the first sentence of the abstract, not the whole paper)
The chat: https://chatgpt.com/share/6a7f1b08-b9dc-83ea-b911-aaf8b65f1c...
>Proofs which fewer and fewer people are able to understand
>Those who do understand, the rate at which models make discoveries exceed the amount of time those humans have in a day
>As a result, we will increasingly use other AI models to validate the proofs AI make for us
I don't see a future where this doesn't apply to everything. Let's say you're an evil CEO, you could ask an AI model to run for days cooking up every possible nefarious scheme to get out of a class-action lawsuit scott-free, to avoid taxes via complex financial engineering, etc. The plans will be far more complex than any human can understand. There simply aren't enough humans with the mental bandwidth to oppose you, so it's just machine vs machine, and if you have more money to pay for more compute, you win.
> > Proofs which fewer and fewer people are able to understand
This is the trend for proofs of famous old theorems in general, even pre-AI. Some particularly extreme examples are the proofs of Fermat's Last Theorem [0], the abc conjecture [1], the classification of finite simple groups [2], and the four colour theorem [3].
[0]: https://en.wikipedia.org/wiki/Wiles's_proof_of_Fermat's_Last...
[1]: https://mathoverflow.net/questions/232087/have-there-been-an...
[2]: https://en.wikipedia.org/wiki/Classification_of_finite_simpl...
[3]: https://en.wikipedia.org/wiki/Four_color_theorem#Proof_by_co...
At the end of the day judge either believes you or not and you are ducked
And if your defense is so complicated then that’s worse for… you
It’s mostly this type of mathematics: classifying what high dimensional spaces can and can’t do.
Maths like this is where the AIs will be most useful in the next few years: tirelessly grinding through every possibility on the edge of human knowledge, filling in gaps and completing maps of no-go regions in the theory space.
Announced by the same people, basically at the same time.
I nearly made it through the first line, OK, sentence of what looks like the Intro, which doesn't bother introducing itself and isn't actually the introduction which occurs later.
I can only describe pages one and two as some sort of foreplay. If you manage to hang on until P108 you can dispense with a safe word in future.
where can one read more about that?
The multiplication they call an "alternating [skew-symmetric?] invariant form": Q0 in symbols I think?
Basically a polynomial function in the components of the "superoctonions" that go into the product.that how I'd vibesplain to a fellow non-pro, anyways
P10, sec 2.4 Lemma 2.8
The complex structure itself is much more complicated.. or else alpoge would have posted a one line verification of the result? But a summary that excites everybody can only be done by a pro...
The next relevant thing in the paper though, that I might be able to attempt to vibesplain using my uh consumer knowhow, is "integrability" (starting in the section labelled "Context")
Or one could just ask Fable/ChatGPT to ELI10
You've got the tool, you've got the will